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Standard Deviation and Variance
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Deviation just means “how far from normal”
Standard Deviation
The Standard Deviation is a measure of how spread out numbers are.
Its symbol is
σ (the Greek letter sigma)
The formula is easy: it is the square root of the Variance. So now
you ask, "What is the Variance?"
Variance
The Variance is defined as:
The average of the squared differences from the Mean.
To calculate the variance follow these steps:
• Work out the Mean (the simple average of the numbers)
• Then for each number: subtract the Mean and square the
result (the squared difference).
• Then work out the average of those squared differences.
These steps are outlined in the handout on data analysis.
Example
You and your friends have just measured the heights of your dogs (in
millimeters):
The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm
and 300mm.
Find out the Mean, the Variance, and the Standard Deviation.
Your first step is to find the Mean:
Answer:
Mean =
600 + 470 + 170 + 430 + 300
_________________________
5
=
1970
____
= 394
5
so the mean (average) height is 394 mm. Let's plot this on a chart.
The red marks are the heights of each dog, the green line is the
mean:
Now, we calculate each dogs difference from the Mean:
To calculate the Variance, take each difference, square it, and then
average the result:
So, the Variance is 21,704.
And the Standard Deviation is just the square root of Variance, so:
Standard Deviation: σ = √21,704 = 147.32... = 147 (to
round up to the nearest mm)
The good thing about the Standard Deviation is that it is useful. Now
we can show which heights are within one Standard Deviation
(147mm) of the Mean (the blue lines):
So, using the Standard Deviation we have a "standard" way of
knowing what is normal, and what is extra large or extra small.
Rottweilers are tall dogs. And Dachshunds are a bit short ... but don't
tell them!
Now we can use standard deviation to get a sense of whether two sets
of data might be significantly different from one another. Let’s
compare the average height and standard deviation of the dogs above
(we’ll call them the Bridlemile dogs) with a group of dogs from kids at
Chapman Elementary School and another group of dogs from
Ainsworth Elementary School:
(1) Bridlemile dogs:
(2) Chapman dogs:
(3) Ainsworth dogs:
394 ± 147 mm
427 ± 162 mm
175 ± 45 mm
If we plot the means on a bar chart, and the standard deviations as
error bars, it would look like this:
700
600
500
400
Series1
300
200
100
0
1
2
3
Using the standard deviation, the range of the data would suggest
that, overall, there is no significant difference between the Bridlemile
and Chapman dogs. Why? Because their standard deviations overlap.
But, there is a significant difference between the Ainsworth dogs and
both the Bridlemile and Chapman dogs, since their standard deviations
do not overlap.
For a tutorial on how to use Microsoft Excel to calculate and plot means and standard
deviations, see: http://www.esf.edu/efb/course/efb226/Excel%20Help.htm